For the other string realization of the 9-dimensional neutral domain wall, namely the \(\overline{D8}/O8\)  stack of a non-supersymmetric type IIA orientifold, a natural interpretation is that instead the cobordism symmetry \(\Omega _1^{\rm Spin}=\mathbb {Z}_2\)  is
broken.
Indeed, since we are not in type I, we cannot make use of the aforementioned relation between cobordism and K-theory and there is no obvious motivation why the cobordism charge should be gauged.
On the other hand, breaking such a symmetry predicts the existence of 7-brane defects which can be nicely identified with the ETW-branes of Local Dynamical Cobordism [1]}, [2]}, [3]}.
In our interpretation, this situation would correspond to Solution II\(^+\) , where in fact the singularities are at finite distance and spacetime closes off.In this picture, it is also tempting to identify the topological \(S^1\)  as the generator of \(\Omega _1^{\rm Spin}\) .
As we have seen, the size of this \(S^1\)  shrinks to zero at the positions of the two ETW-branes and thus the boundary conditions for the fermions should be periodic.
